Did Frank Sinatra ever record I Left My Heart in San Francisco?

Yes.

Frank Sinatra recorded "I Left My Heart In San Francisco" on August 27 1962. The arrangement was by Neal Hefti and the orchestra was conducted by Nelson Riddle. The single was coupled with "The Look Of Love". It was Master Number 1427 and the Single Number was Reprise 20.107. It was released in Italy on album number FS5238. The song is now available on the Box Set "The Complete Reprise Studio Recordings".

"I Left My Heart In San Fransisco" was withdrawn when Frank Sinatra recorded the song and one day, while he was listening to his recording, realized that, compared to Tony Bennett's version, his version was quite bad. As a result of this, he ordered the record to be withdrawn.